Skip to main content

Minimal Versus Orthogonal Projections onto Hyperplanes in \(\ell _1^{n}\) and \(\ell _{\infty }^{n}\)

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 83))

Abstract

In this paper, we explore the relation between the minimal and the orthogonal projections onto hyperplanes in \(\ell _1^n\) and \(\ell _\infty ^n.\)

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Blatter, J., Cheney, E.W.: Minimal projections on hyperplanes in sequence spaces. Ann. Mat. Pura Appl. 101(4), 215–227 (1974). MR0358179 (50 #10644)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chalmers, B.L., Lewicki, G.: Symmetric spaces with maximal projection constants. J. Funct. Anal. 200(1), 1–22 (2003). MR1974085 (2004b:46009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chalmers, B.L., Lewicki, G.: A proof of the Grünbaum conjecture. Studia Math. 200(2), 103–129 (2010). MR2725896 (2011j:46004)

    Google Scholar 

  4. Chalmers, B.L., Metcalf, F.T.: The determination of minimal projections and extensions in \(l^1\). Trans. Amer. Math. Soc. 329(1), 289–305 (1992). MR1034660 (92e:41017)

    MATH  MathSciNet  Google Scholar 

  5. Grünbaum, B.: Projection constants. Trans. Amer. Math. Soc. 95, 451–465 (1960). MR0114110 (22 #4937)

    Article  MATH  MathSciNet  Google Scholar 

  6. Odyniec, W., Lewicki, G.: Minimal projections in Banach spaces, Lecture Notes in Mathematics. Problems of Existence and Uniqueness and Their Application, vol. 1449. Springer, Berlin (1990). MR1079547 (92a:41021)

    Google Scholar 

  7. Wojtaszczyk, P.: Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, vol. 25. Cambridge University Press, Cambridge (1991). MR1144277 (93d:46001)

    Book  Google Scholar 

  8. Lewicki, G., Prophet, M.: Codimension-one minimal projections onto Haar subspaces. J. Approx. Theory 127, 198–206 (2004). MR2058158 (2005h:41070)

    Article  MATH  MathSciNet  Google Scholar 

  9. Isbell, J.R., Semadeni, Z.: Projection constants and spaces of continuous functions. Trans. Amer. Math. Soc. 107, 38–48 (1963). MR0146649 (26 #4169)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chalmers, B.L., Shekhtman, B.: On minimal and almost locally minimal and orthogonal minimal projections. In: Trends in Approximation Theory (Nashville, TN, 2000), Innovations in Applied Mathematics, pp. 49–52. Vanderbilt University Press, Nashville (2001). MR1937998

    Google Scholar 

  11. Chalmers, B.L., Shekhtman, B.: On spaces admitting minimal projections which are orthogonal. In: Approximation Theory X (St. Louis, MO, 2001), Innovations in Applied Mathematics, pp. 113–116. Vanderbilt University Press, Nashville (2002). MR1924853 (2003f:41045)

    Google Scholar 

  12. Shekhtman, B.: Some examples concerning projection constants, approximation theory, spline functions and applications, (Maratea, 1991). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 356, 471–476 (1992). MR1165993 (93f:41038)

    MathSciNet  Google Scholar 

  13. Zippin, M.: Orthogonal almost locally minimal projections on \({\ell }^n_1\). Israel J. Math. 115, 253–268 (2000). MR1749681 (2001h:46022)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kadec’, M.I., Snobar, M.G.: Certain functionals on the Minkowski compactum. Mat. Zametki 10, 453–457 (1971). MR0291770 (45 #861)

    MathSciNet  Google Scholar 

  15. Bohnenblust, F.: Convex regions and projections in Minkowski spaces. Ann. of Math. 39, 301–308 (1938)

    Google Scholar 

  16. Franchetti, C.: Projections onto hyperplanes in Banach spaces. J. Approx. Theory 38(4), 319–333 (1983). MR711458 (84h:46023)

    Article  MATH  MathSciNet  Google Scholar 

  17. Franchetti, C.: The norm of the minimal projection onto hyperplanes in \(L^p[0,1]\) and the radial constant. Boll. Un. Mat. Ital. B 7(4), 803–821 (1990). MR1086705 (92e:46029)

    MathSciNet  Google Scholar 

  18. Rolewicz, S.: On projections on subspaces of codimension one. Studia Math. 96(1), 17–19 (1990). MR1055074 (91i:46017)

    MATH  MathSciNet  Google Scholar 

  19. Skrzypek, L.: On the \(L_p\) norm of the Rademacher projection and related inequalities. Proc. Amer. Math. Soc. 137(8), 2661–2669 (2009). MR2497479 (2010d:41043)

    Article  MATH  MathSciNet  Google Scholar 

  20. Skrzypek, L.: Chalmers-Metcalf operator and uniqueness of minimal projections in \(\ell _\infty ^n\) and \(\ell _1^n\) spaces. In: Neamtu, M., Schumaker L. (eds.) Springer Proceedings in Mathematics, Approximation Theory XIII: San Antonio 2010, vol. 13, pp. 331–344 (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Boris Shekhtman .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Shekhtman, B., Skrzypek, L. (2014). Minimal Versus Orthogonal Projections onto Hyperplanes in \(\ell _1^{n}\) and \(\ell _{\infty }^{n}\) . In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_20

Download citation

Publish with us

Policies and ethics